List of top Mathematics Questions asked in JEE Advanced

Let i^,j^\hat{i}, \hat{j} and k^\hat{k} be the unit vectors along the three positive coordinate axes Let a→=3i^+j^−k^, \overrightarrow{ a }=3 \hat{ i }+\hat{ j }-\hat{ k }, b→=i^+b2j^+b3k^,b2,b3∈R,\overrightarrow{ b }=\hat{ i }+ b _2 \hat{ j }+ b _3 \hat{ k }, b _2, b _3 \in R , c→=c1i^+c2j^+c3k^,c1,c2,c3∈R \overrightarrow{ c }= c _1 \hat{ i }+ c _2 \hat{ j }+ c _3 \hat{ k }, c _1, c _2, c _3 \in R be three vectors such that b2b3>0,a⃗⋅b⃗=0b_2 b_3>0, \vec{a} \cdot \vec{b}=0 and (0−c3c2c30−c1−c2c10)(1b2b3)=(3−c11−c2−1−c3)\begin{pmatrix}0 & -c_3 & c_2 \\c_3 & 0 & -c_1 \\-c_2 & c_1 & 0\end{pmatrix}\begin{pmatrix} 1 \\b_2 \\b_3\end{pmatrix}=\begin{pmatrix}3-c_1 \\1-c_2 \\-1-c_3\end{pmatrix} . Then, which of the following is/are TRUE?
Let GG be a circle of radius R>0R>0. Let G1,G2,…,GnG _1, G _2, \ldots, G _{ n } be nn circles of equal radius r>0r>0. Suppose each of the nn circles G1,G2,…,GnG _1, G _2, \ldots, G _{ n } touches the circle GG externally. Also, for i=1,2,…,n−1i =1,2, \ldots, n -1, the circle GiG _{ i } touches Gi+1G _{ i +1} externally, and GnG _{ n } touches G1G _1 externally. Then, which of the following statements is/are TRUE?
Consider the hyperbola x2100−y264=1\frac{x^2}{100}-\frac{y^2}{64}=1 with foci at SS and S1S_1, where SS lies on the positive xx-axis Let PP be a point on the hyperbola, in the first quadrant Let ∠SPS⁡1=α\angle \operatorname{SPS}_1=\alpha, with α<π2\alpha<\frac{\pi}{2}. The straight line passing through the point SS and having the same slope as that of the tangent at PP to the hyperbola, intersects the straight line S1PS _1 P at P1P _1 Let δ\delta be the distance of PP from the straight line SP1SP _1, and β=S1P\beta= S _1 P. Then the greatest integer less than or equal to βδ9sin⁡α2\frac{\beta \delta}{9} \sin \frac{\alpha}{2} is _____.
Let zˉ\bar{z} denote the complex conjugate of a complex number zz. If zz is a non-zero complex number for which both real and imaginary parts of (zˉ)2+1z2(\bar{z})^2+\frac{1}{ z ^2} are integers, then which of the following is/are possible value(s) of ∣z∣|z|?
Consider the functions f,g:R→Rf, g: R \rightarrow R defined by
f(x)=x2+512 and g(x)={2(1−4∣x∣3),∣x∣≤34,0,∣x∣>34.f(x)=x^2+\frac{5}{12} \text { and } g(x)= \begin{cases} 2\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4}, \\0, & |x|>\frac{3}{4}. \end{cases}
If α\alpha is the area of the region {(x,y)∈R×R:∣x∣≤34,0≤y≤min⁡{f(x),g(x)}},\left\{( x , y ) \in R \times R :| x | \leq \frac{3}{4}, 0 \leq y \leq \min \{f( x ), g( x )\}\right\}, then the value of 9α9 \alpha is ____.
Let β\beta be a real number. Consider the matrix
A=(β0121−231−2)A=\begin{pmatrix}\beta & 0 & 1 \\2 & 1 & -2 \\3 & 1 & -2\end{pmatrix}
If A7−(β−1)A6−βA5A^7-(\beta-1) A^6-\beta A^5 is a singular matrix, then the value of 9β9 \beta is __.
Let l1,l2,…,l100l_1, l_2, \ldots, l_{100} be consecutive terms of an arithmetic progression with common difference d1d_1, and let w1,w2,…,w100w_1, w_2, \ldots, w_{100} be consecutive terms of another arithmetic progression with common difference d2d_2, where d1d2=10d_1 d_2=10 For each i=1,2,…,100i=1,2, \ldots, 100, let RlR_l be a rectangle with length lil_i, width wiw_i and area AiA i If A51−A50=1000A_{51}-A_{50}=1000, then the value of A100−A90A_{100}-A_{90} is ___

Let zˉ\bar{z} denote the complex conjugate of a complex number zz and let i=−1i=\sqrt{-1} In the set of complex numbers, the number of distinct roots of the equation zˉ−z2=i(zˉ+z2)\bar{z}-z^2=i\left(\bar{z}+z^2\right) is ___

Let P1P_1 and P2P_2 be two planes given by P1:10x+15y+12z−60=0, P_1: 10 x+15 y+12 z-60=0, P2:−2x+5y+4z−20=0 P_2:-2 x+5 y+4 z-20=0 Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P1P_1 and P2P_2 ?
Consider the equation ∫1e(log⁡ex)1/2x(a−(log⁡ex)3/2)2dx=1,a∈(−∞,0)∪(1,∞)\int\limits_1^e \frac{\left(\log _{ e } x \right)^{1 / 2}}{x\left(a-\left(\log _{ e } x \right)^{3 / 2}\right)^2} dx =1, \quad a \in(-\infty, 0) \cup(1, \infty) Which of the following statements is/are TRUE ?

Let ABCA B C be the triangle with AB=1,AC=3A B=1, A C=3 and ∠BAC=π2\angle B A C=\frac{\pi}{2} If a circle of radius r>0r>0 touches the sides AB,ACA B, A C and also touches internally the circumcircle of the triangle ABCA B C, then the value of rr ___

Let zˉ\bar{z} denote the complex conjugate of a complex number zz and let i=−1i=\sqrt{-1} In the set of complex numbers, the number of distinct roots of the equation zˉ−z2=i(zˉ+z2)\bar{z}-z^2=i\left(\bar{z}+z^2\right) is ___
Let l1,l2,…,l100l_1, l_2, \ldots, l_{100} be consecutive terms of an arithmetic progression with common difference d1d_1, and let w1,w2,…,w100w_1, w_2, \ldots, w_{100} be consecutive terms of another arithmetic progression with common difference d2d_2, where d1d2=10d_1 d_2=10 For each i=1,2,…,100i=1,2, \ldots, 100, let RlR_l be a rectangle with length lil_i, width wiw_i and area AiA i If A51−A50=1000A_{51}-A_{50}=1000, then the value of A100−A90A_{100}-A_{90} is ___
Consider the parabola y2=4xy^2=4 x Let SS be the focus of the parabola A pair of tangents drawn to the parabola from the point P=(−2,1)P=(-2,1) meet the parabola at P1P_1 and P2P_2 Let Q1Q_1 and Q2Q_2 be points on the lines SP1S P_1 and SP2S P_2 respectively such that PQ1P Q_1 is perpendicular to SP1S P_1 and PQ2P Q_2 is perpendicular to SP2S P_2 Then, which of the following is/are TRUE ?
Let PQRSPQRS be a quadrilateral in a plane, where QR=1,∠PQR=∠QRS=70∘,∠PQS=15∘QR =1, \angle PQR =\angle QRS =70^{\circ}, \angle PQS =15^{\circ} and ∠PRS=40∘\angle PRS =40^{\circ}. If ∠RPS=θ∘,PQ=α\angle RPS =\theta^{\circ}, PQ =\alpha and PS=βPS =\beta, then the interval(s) that contain(s) the value of 4αβsin⁡θ∘4 \alpha \beta \sin \theta^{\circ} is/are
If y(x)y(x) is the solution of the differential equation xdy−(y2−4y)dx=0 for x>0,y(1)=2,x d y-\left(y^2-4 y\right) d x=0 \text { for } x>0, y(1)=2, and the slope of the curve y=y(x)y=y(x) is never zero, then the value of 10y(2)10 y(\sqrt{2}) is ____.
Considering only the principal values of the inverse trigonometric functions, the value of 32cos⁡−122+π2+14sin⁡−122π2+π2+tan⁡−12π\frac{3}{2} \cos ^{-1} \sqrt{\frac{2}{2+\pi^2}}+\frac{1}{4} \sin ^{-1} \frac{2 \sqrt{2} \pi}{2+\pi^2}+\tan ^{-1} \frac{\sqrt{2}}{\pi} is ___
In a triangle ABCABC, let AB=23,BC=3AB =\sqrt{23}, BC =3 and CA=4CA =4. Then the value of cot⁡A+cot⁡Ccot⁡B\frac{\cot A +\cot C }{\cot B } is ______
Let u⃗,v⃗\vec{u}, \vec{v} and w⃗\vec{w} be vectors in three-dimensional space, where u⃗\vec{u} and v⃗\vec{v} are unit vectors which are not perpendicular to each other and
u⃗⋅w⃗=1,v⃗⋅w⃗=1,w⃗⋅w⃗=4\vec{ u } \cdot \vec{ w }=1, \vec{ v } \cdot \vec{ w }=1, \vec{ w } \cdot \vec{ w }=4
If the volume of the parallelepiped, whose adjacent sides are represented by the vectors u⃗,v⃗\vec{ u }, \vec{ v } and w⃗\vec{ w } is 2\sqrt{2} , then the value of ∣3u⃗+5v⃗∣|3 \vec{u}+5 \vec{v}| is ______
Let 
S1={(i,j,k):i,j,k∈{1,2,…,10}},S_{1}=\{( i , j , k ): i , j , k \in\{1,2, \ldots , 10\}\}, 
S2={(i,j):1≤i<j+2≤10,i,j∈{1,2,…10}}, S_{2}=\{( i , j ): 1 \leq i < j +2 \leq 10, i , j \in\{1,2, \ldots 10\}\}, 
S3={(i,j,k,l):1≤i<j<k<l,i,j,k,l∈{1,2,…,10}} S_{3}=\{( i , j , k , l ): 1 \leq i < j < k < l , i , j , k , l \in\{1,2, \ldots , 10\}\} and 
S4={(i,j,k,l):i,j,k S_{4}=\{( i , j , k , l ): i , j , k and l l are distinct elements in {1,2,…,10}}\{1,2, \ldots , 10\}\}.
 If the total number of elements in the set SrS_{ r } is nr,r=1,2,3,4n_{ r }, r =1,2,3,4, then which of the following statements is(are) TRUE?
Let f:R→Rf: R \rightarrow R be defined by f(x)=x2−3x−6x2+2x+4f(x)=\frac{x^{2}-3 x-6}{x^{2}+2 x+4}. Then which of the following statements is(are) TRUE?
The area of the region {(x,y):0≤x≤94,0≤y≤1,x≥3y,x+y≥2}\{ {(x, y): 0 ≤ x ≤ \frac{9}{4}, 0 ≤ y ≤ 1, x ≥ 3y, x + y ≥ 2}\}
Let θ1, θ2,…., θ10 \theta_1,  \theta_2, ….,  \theta_{10} be positive valued angles (in radian) such that θ1+ θ2+….+ θ10=2π \theta_1+  \theta_2+ ….+  \theta_{10} = 2\pi. Define the complex numbers z1=𝑒𝑖θ1,𝑧𝑘=𝑧𝑘−1𝑒𝑖θkfor  k=2,3,…,10,z_1 = 𝑒^{ 𝑖\theta_1}, 𝑧_𝑘 = 𝑧_{𝑘−1}𝑒^ {𝑖\theta_k} \text{for}\  k = 2, 3, …, 10, where i=−1i = \sqrt{-1}. Consider the statement P and Q given below:
 P:∣z2−z1∣+∣z3−z2∣+…+∣z10−z9∣+∣z1−z10∣≤2πP: \left|z_2 - z_1\right| + \left|z_3 - z_2\right| + \ldots + \left|z_{10} - z_9\right| + \left|z_1 - z_{10}\right| \leq 2\pi
Q:∣z22−z12∣+∣z32−z22∣+…+∣z102−z92∣+∣z12−z102∣≤4πQ: \left|z_{22} - z_{12}\right| + \left|z_{32} - z_{22}\right| + \ldots + \left|z_{102} - z_{92}\right| + \left|z_{12} - z_{102}\right| \leq 4\pi
Then,
For any positive integer nn, let Sn:(0,∞)→RS_{n}:(0, \infty) \rightarrow R be defined by
Sn(x)=∑k=1ncot⁡−1(1+k(k+1)x2x),S_{n}(x)=\displaystyle\sum_{k=1}^{n} \cot ^{-1}\left(\frac{1+k(k+1) x^{2}}{x}\right),
where for any x∈R,cot⁡−1(x)∈(0,π)x \in R, \cot ^{-1}(x) \in(0, \pi) and tan⁡−1(x)∈(−π2,π2)\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right). Then which of the following statements is(are) TRUE?
For any 3×33 \times 3 matrix MM, let ∣M∣| M | denote the determinant of MM. Let
E=[12323481318],P=[100001010]E=\begin{bmatrix}1 & 2 & 3 \\2 & 3 & 4 \\8 & 13 & 18\end{bmatrix}, P=\begin{bmatrix} 1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix} and F=[13281813243]F=\begin{bmatrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{bmatrix}
If QQ is a non-singular matrix of order 3×33 \times 3, then which of the following statements is(are) TRUE?